Insights into the Impact of Surface Hydrophobicity on Droplet

Subscriber access provided by CORNELL UNIVERSITY LIBRARY

Article

Insights into the Impact of Surface Hydrophobicity on Droplet Coalescence and Jumping Dynamics Hongxia Li, Weilin Yang, Abulimiti Aili, and TieJun Zhang Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b02146 • Publication Date (Web): 02 Aug 2017 Downloaded from http://pubs.acs.org on August 3, 2017

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

Langmuir is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 41

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Insights into the Impact of Surface Hydrophobicity on Droplet Coalescence and Jumping Dynamics Hongxia Li, Weilin Yang, Abulimiti Aili, TieJun Zhang* Department of Mechanical and Materials Engineering, Masdar Institute, Khalifa University of Science and Technology, P.O. Box 54224, Abu Dhabi, UAE *Email: [email protected]

ABSTRACT Droplet coalescence jumping on superhydrophobic surfaces attracts much research attention owing to its capability in enhancing condensation for energy and water applications. In this work, we reveal the impact of the finite surface adhesion to explain velocity discrepancies observed in recent droplet jumping studies, particularly when droplet sizes are a few micrometers (1–10 µm). Surface adhesion, which is usually neglected, can significantly affect both droplet coalescence and departure dynamics. It causes oscillations on velocity and contact area in the droplet coalescence process, as observed numerically and experimentally. Comparing the increasing rate of jumping velocity with contact angle for three different droplet sizes, we show that smaller droplets exhibit higher sensitivity to the change of surface hydrophobicity. We also specify the range of surface superhydrophobicity where the jumping velocity monotonically decreases (θ≳170°), or increases (θ≲160°), or changes nonmonotonically in transition (160°≲θ ≲170°) with droplet size. As a result, there exists a broad jumping velocity range for micrometer-sized droplets on a superhydrophobic surface 1 ACS Paragon Plus Environment

Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 41

with a slight contact angle variation. This work offers an extended understanding of the droplet coalescence and jumping dynamics to resolve the discrepancies in recent experimental observations. KEYWORDS: Hydrophobicity, Coalescence, Droplet jumping, Oscillation, Lattice Boltzmann modeling

1. INTRODUCTION Coalescence-induced jumping of droplets on superhydrophobic surfaces is an interesting phenomenon1–7 where two or more droplets coalesce and the resulting droplet jumps away from the surface. Droplet jumping has found applications in condensation heat transfer,1,3,8–15 selfcleaning surfaces,16,17 anti-icing,18,19 water harvesting,20 electrostatic energy harvesting21, etc. Superhydrophobicity arises from the combination of low surface energy and surface structures.22–26 To enhance the superhydrophobicity, the introduction of nano-/micro structures on surfaces has been widely used.27–29 Different fabrication techniques such as chemical etching,5,10,30,31 anodization,32,33 electrochemical deposition,34 and physical vapor deposition35 may produce various degrees of macroscopic superhydrophobicity in terms of static contact angle and contact angle hysteresis (CAH). Specifically, the contact angle becomes smaller when droplet size is smaller than or comparable to surface structure.36–38 From this point of view, the influence of surface superhydrophobicity coupled with the droplet size on droplet jumping is worth further study, instead of simply using a single contact angle or neglecting the surface adhesion.

2 ACS Paragon Plus Environment

Page 3 of 41

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Researchers have conducted many studies on droplet jumping on superhydrophobic surfaces.1– 4,7,39–45

When inertia dominates the jumping dynamics, the jumping velocity of two coalesced

droplets with an equal radius follows an inertial-capillary scaling law1 U∝

γ lv / ( ρ R ) ,

(1)

ρ R 3 / γ lv .

(2)

and the droplet coalescence time scale τ is τ∝

When the droplet becomes smaller, the deceasing Ohnesorge number ( Oh = µ / (ρσ R)1/2 ) indicates that viscous effects become increasingly important.39–47 The viscous dissipation can lead to a decrease in jumping velocity. A number of studies have reported that coalescing water droplets do not jump below a critical radius ~ 10 µm (Oh~0.1) due to viscous dissipation.39–41,46,47 However, jumping of even smaller droplets have been reported recently. For instance, jumping of 10 µm droplets with an apparently high speed of 1.4 m/s was observed by R. Enright et al.2 as well as jumping of 5 µm droplets by H. Cha et al.4 These observations imply that the critical radius 10 µm for jumping is obviously inaccurate and that the dominant role of viscous dissipation is questioned. Also, F. Liu et al.42,43 pointed out that it may be the finite surface adhesion that leads to a much larger threshold radius1 on an actual nanostructured superhydrophobic surface, while the numerically obtained cutoff radius for the Leidenfrost surface can be as small as ~0.3 µm without any droplet-surface adhesion. Later, H. Cha et.44 observed the minimum droplet departure with radii of 700 nm, 4.5 µm, 20 µm, and no jumping on four CNT-coated surfaces with receding contact angle of 164°, 152°, 146°, and 140°, respectively. These observations show that the droplet size threshold for jumping closely depends on the surface hydrophobicity levels. Through analyzing the coalescence time scales 3 ACS Paragon Plus Environment

Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 4 of 41

experimentally and theoretically, they also showed that inertial-capillary dynamics still govern jumping down to nanodroplet length scales before the viscous effect becomes dominant. Meanwhile, Z. Liang and P. Keblinski45 numerically verified inertial-capillary scaling of jumping velocities for nanodroplet through Molecular Dynamics modeling. Therefore, the underlying cause preventing micrometer-sized droplet jumping on actual surfaces is not viscous dissipation but most likely the finite surface adhesion. Discrepancies are also found in the magnitude and evolution of droplet jumping velocity with droplet size. For a droplet on a less superhydrophobic surface1, its jumping velocity was observed to reach a peak and then decrease with increasing droplet size, and the velocity magnitude was below 1.0 m/s. However, on a surface with higher contact angle of 170.2°±2.4°,2 the measured highest jumping velocity was 3.5 m/s about four times higher than Ref.1 And the overall jumping velocity decreased with the droplet size. M. K. Kim et al.3 observed the same trend over the measured range (5—50 µm) with a macroscopic contact angle of 170.5°±7.2°. Their results agree well with the recent report by H. Cha et al.4 These five experimental results are reconstructed in Fig. 1, showing a broad measured velocity range especially at droplet sizes smaller than 20 µm. Even in the same study, e.g. in Ref.2, the jumping velocity widely ranges from 0.7 to 3.5 m/s for the specific small droplet size of 5 µm. Interestingly, consistency can be found between all surfaces for relatively larger droplets. All the above discussions imply that the slight variation in the contact angle of small droplets is responsible for their wide jumping velocity range. It indicates that finite surface adhesion can significantly affect the jumping dynamics. However, a limited number of studies have been conducted to investigate how sensitive micrometer-sized droplets are to a finite variation in superhydrophobicity.44,48 Through analyzing jumping velocities of different droplet sizes for 4 ACS Paragon Plus Environment

Page 5 of 41

three different contact angles, 150°, 160° and 180°, Cheng et al.48 found that the contact angle has significant effect on the reduction of droplet jumping velocity. Since the fabrication of superhydrophobic surfaces with continuously varying contact angle is challenging, numerical simulation comes into play. Different numerical methods have been successfully applied to droplet jumping studies, like finite-element-based computational method (FEM),2,42,48 lattice Boltzmann method (LBM)40,49–51 and molecular dynamics simulation (MD).45 Among those numerical studies, LBM is one of the most popular simulation methods. Originating from the kinetic Boltzmann Transport Equation, LBM provides a different way to simulate the microscale multiphase flow and the interface evolution based on direct particle interaction.52 In droplet jumping problems, velocity and pressure field evolution inside the droplet can be observed through LBM. Most importantly, it is capable of simulating multiphase flow under various surface wettability conditions, which makes it suitable for studying the influence of surface adhesion during the droplet jumping process. 4

jumping velocity ( m/s )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Experimental data points low hydrophobicity

3

J. B. Boreyko et. al [1]

high hydrophobicity R. Enright et. al [2] R. Enright et.al [2] M. K. Kim et. al [3] H Cha et. al [4]

2

1

0 0

20

40

60

80

100

droplet size ( µm )

FIG.1. Jumping velocity vs. droplet radius (reproduced from experimental results in references.14

)


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 41

In order to study the sensitivity of jumping droplets to the surface hydrophobicity, it is important to have a good understanding of the droplet coalescence and departure dynamics. Here, we first simulate the coalescence-induced jumping process using LBM approach. The momentum evolution and streamlines inside the coalescing droplets are analyzed and compared with data from open literature. In droplet coalescence process, oscillations on droplet velocity and liquidsolid contact area are observed in simulation and demonstrated experimentally. Later, we look into the droplet departure process and compare the jumping velocities of several different droplet sizes (1–10 µm) over a large contact angle range (140–180˚). The forces exerted on the departing droplet are analyzed to understand the simulated jumping dynamics. Finally, the jumping velocity results as well as the minimum departure droplet sizes are summarized in cases with various levels of surface hydrophobicity. 2. RESULTS AND DISCUSSION The Shan-Chen type of LBM model53 incorporated with the Rothman-Keller equation of state (EOS) model54–56 is used for droplet jumping dynamics modeling. The model description is provided in our previous work.57,58 In this study, T/Tc is set as 0.85. The liquid density is 6.1 with vapor density 0.5, and surface tension is 1.6. By simultaneously changing the droplet size and calculation domain with a fixed droplet-domain size ratio, we simulated cases of Oh=0.05 ~0.2. Gravity is neglected since the Bond number is between 10-6~10-4 in all simulations. The contact angle in all the simulations ranges from 140° to 180°, and the realization method is described in reference papers.52,59 More details about parameter settings are given in Supporting Information. 2.1 Droplet Coalescence and Jumping Dynamics


Page 7 of 41

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Droplet coalescence jumping is a process during which the unbalanced capillary pressure leads to liquid movement and curvature evolution (Fig. 2). In this process, the excess surface free energy is released and converted into the kinetic energy of the jumping droplet. Once the coalescence starts, the negative curvature at the contact point of the two coalescing droplets results in a lowest pressure point, which is even lower than the ambient pressure. The pressure gradient then leads to liquid movement inside the droplet. Figure 2 shows the streamlines and momentum distribution inside the droplets at the very beginning of coalescence. The color bar indicates the momentum magnitude and direction. As seen from the flow field and streamlines, the pressure gradient drives the liquid to flow towards the coalescence point, due to the negative curvature at the coalescence bridge. The results are further compared with the conventional finite-element-based numerical method2 (the inset in Fig. 2), showing a good agreement at this stage of the coalescence jumping process. In the following part, we show that the LBM simulation enables us to scrutinize the influence of surface adhesion in the droplet coalescence and departure process, respectively.


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 41

FIG. 2. Momentum and streamlines at the early stage of coalescence jumping on a superhydrophobic surface. The lattice Boltzmann simulation result (bottom) is compared with finite-element-based simulation results from Ref.2 The color bar shows the momentum magnitude and direction, with the blue color representing positive (upward) and the yellow color representing negative (downward).

The whole jumping process can be divided into three phases. In Phase I (coalescence), the surface free energy is converted into the kinetic energy and the surface curvature goes through a major evolution. In Phase II (departure), the coalesced droplet overcomes the adhesion force and departs from the solid surface. In Phase III, the coalesced droplet jumps away from the surface and its velocity reduces to zero due to drug force. In Fig. 3, we plot the coalescence jumping dynamics of a droplet on a moderate superhydrophobic surface (contact angle ~ 167°). The droplet size is characterized with the Ohnesorge number Oh = µ / (ρσ R)1/2 . Since the viscosity µ, density ρ, and surface tension σ are constant once the thermal-physical properties of water are fixed, low Oh numbers represent large droplet sizes. In this figure, the droplet diameter is around 1.5 µm. Given that Bond number Bo = ρ gR2 / σ ≈ 10−6

1 , the gravity effect is negligible in the

simulation. The velocity ν in the y+ direction is defined as the average velocity based on the mass center of the merged droplet, that is

vy + =

∫ v dΩ Ω y

Ω

,

(3)

where Ω is the droplet volume, y is the direction normal to the solid surface. At the beginning of Phase I, the total momentum direction is negative. Once the bottom part of the coalesced bridge


Page 9 of 41

contacts the substrate surface and bounces back, y-component momentum towards the surface reverses its direction, thus leading to an abrupt increase of νy+ (at around 50 µs in Fig. 3). Because the surface energy is continuously converted into kinetic energy, νy+ keeps increasing until it reaches the maximum point, denoted as νy+,max. Its value is directly related to the released surface free energy. Interestingly, oscillations of the y-component of velocity νy+ are observed in this process. At Phase II, the droplet has to overcome liquid-solid adhesion to jump away from the solid surface. νy+ decreases with a constant deceleration rate, which is related to the surface adhesion force exerted on the droplet. How the surface wettability affects this deceleration process is the focus of this work and is discussed later. 8

6

Phase I coalescence

4

e as le re

+

y veloctiy ν y (m/s)

2

su

ce rfa

en

r ge

y

Phase II departure ad h esi

on

ν

y+,max

Phase III floating

res is

tan

ce

air friction

νj

0 jumping

+

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

-2 coalescence 0

50

o

Contact angle 167 , Oh=0.1

100

150

200

250

Jumping process (µs)

FIG. 3. The evolution of the y-component velocity and droplet shape during the droplet coalescence jumping process simulated by LBM. The whole process is divided into three phases. In Phase I (coalescence), the surface curvature goes through a major evolution and the surface free energy is partially converted into kinetic energy. In Phase II (departure), the droplet overcomes the surface adhesion and departs from the solid surface. In Phase III, the drug force slows down the jumping droplet.


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 41

2.2 Oscillations during Droplet Coalescence Oscillations of νy+ occur in Phase I (Fig. 3) after the liquid bridge impinges on the solid substrate. Oscillations of the contact area exist as well in Fig. 4(a). In order to explain this phenomenon, we visualize the liquid-vapor interface curvature evolution and streamlines inside the droplet in Fig. 4(b). At State S1, the streamlines show that the liquid flows from the droplet bottom to the coalescence bridge where the lowest pressure is. As a result, the contact area decreases at the very beginning. As the coalescence bridge moves towards solid substrate, the contact area becomes larger gradually. At State S2, the contact area reaches its maximum when the coalescence bridge touches the substrate surface. However, the contact area decreases again due to the liquid velocity bounce back. To be clear, it can be found that the liquid flow direction is reversed after the coalescence bridge impinged on the substrate by comparing the streamlines in S2 and S3. The liquid interface starts to bounce back and moves upward until it reaches the highest point. Eventually, it falls down again due to the negative capillary pressure caused by interface curvature as shown in S4.


Page 11 of 41

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

FIG. 4. Contact area evolution during the coalescence jumping process. (a) Normalized contact area. At State S1, the contact area decreases first due to liquid flow from the droplet bottom towards the coalescence bridge. At S2, the liquid-vapor interface fully touches the solid surface. From S2 to S3, the downward flow bounces back and leads to the decrease of contact area. (b) Corresponding droplet morphologies and streamlines of the four states in (a).

FIG. 5. Coalescence of two droplets pinned on needle tips. Oscillation of the liquid-vapor interface occurs due to the pinning effect and surface tension. The arrows are showing the liquidvapor interface moving direction.


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 41

The oscillation process was also experimentally observed when two pinned droplets were coalescing (Fig. 5). Two droplets made by the syringe needle were brought together and coalesced at t=0.0 ms. After 4.6 ms, the liquid bridge reached the lowest point. As illustrated in Fig. 5, the moving direction reverses due to the vertical component of surface tension even without the assistance of the upward force resulting from the impingement with solid surface. Because droplets were pinned at the needle tip, the interface moved downward again at t=4.9 ms. Therefore, oscillation started. More experimental details can be found in the Supporting Information. Similar bouncing process, termed as pancake bouncing, that occurs at superhydrophobic surfaces has also been reported and discussed.60–63 Compared with pancake bouncing, oscillations arise with higher possibility in droplet coalescence process. In pancake bouncing process, it is the whole droplet that impinges on the substrate; thus the fluid first spreads, retracts, and then bounces. However, in droplet coalescence, only the bottom part of liquid bridge impinges the substrate. Since the top part of droplet has upward moving velocity and cannot push the liquid underneath, the liquid bridge can directly bounce back rather than spreading. Oscillations repeat due to the pinning of contact line (from surface adhesion) and gradually disappear due to the liquid-solid interaction and viscous dissipation. In the Supporting Information, we simulated the droplet coalescence dynamics at different contact angles. In fact, oscillation does not exist anymore when the contact angle is near 180°, which implies, from one aspect, that surface adhesion has important influence on jumping dynamics.

2.3 Dependence of Jumping Velocity on Droplet Size and Surface Hydrophobicity In order to demonstrate the effect of surface adhesion on different size droplets at different surface hydrophobicity levels, we plot the change of νy+ during the coalescence jumping process 12 ACS Paragon Plus Environment

Page 13 of 41

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

for Oh = 0.1 and 0.2 with contact angles of 165° and 175°, as shown in Fig. 6. Here, νj,s1, νj,s2 are the jumping velocities of a smaller droplet (Oh=0.2) at contact angles 175° and 165°, respectively. Similarly, νj,l1 and νj,l2 are for the droplet with Oh=0.1. As mentioned, our focus is the effect of surface adhesion resistance on droplet velocity deceleration at Phase II. For a specific droplet size, e.g. Oh=0.2 (the dashed red and black curves), the velocity reduction slope is similar for two different contact angles. However, the time-span differs with the contact angle. The deceleration time for the contact angle of 175° is much shorter, so that the corresponding jumping velocity νj,s1 is larger than νj,s2. Large contact angle leads to small jumping velocity. However, if we compare the dashed and solid curves, different slopes can be found for different droplet sizes, implying that the deceleration magnitude is a function of the droplet size. The velocity of the small droplet (dashed curves) decreases faster than the large droplet. The reason lies in the forces exerted on the droplet during the departure process as illustrated later. Another piece of information from Fig. 6 is the change in jumping velocity when we change the surface hydrophobicity for different droplet sizes. Let us define the jumping velocity drop as ∆νj,s= νj,s1 - νj,s2 under the same contact angle change, indicated by the horizontal blue and purple lines. As can be seen, the droplet jumping velocity drop for Oh=0.2 is larger than that for Oh=0.1. In other words, the smaller size droplet is more sensitive to the surface hydrophobicity change. It can be further inferred from the above that the jumping velocity distribution of small droplet sizes reflects the microscale homogeneity of surface hydrophobicity. From this point of view, we can explain that the wide jumping velocity distributions at small droplet sizes, found in almost all experiments in Ref. 2-4, are due to non-uniform surface hydrophobicity and the higher sensitivity of smaller droplets to the non-uniformity.


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 41

FIG. 6. The effect of surface wettability on the jumping velocity of different droplet sizes. νj,s1 , νj,s2 , νj,l1 and νj,l2 indicate the jumping velocities of smaller (Oh = 0.2) and larger (Oh = 0.1) droplets with contact angles of 175° and 165°, respectively. When the contact angle changes from 175° to 165 °, the jumping velocity drop ∆νj,s of the smaller droplet (Oh = 0.2, dashed curve) is larger than that of the larger droplet (Oh = 0.1, solid curve).

FIG. 7. Forces acting on a liquid droplet due to liquid-solid adhesion. At equilibrium state (blue colored droplet), in normal direction to the solid surface, the adhesion force exerted by solid-ondroplet at the contact line γlvsinθ is balanced by the repelling force ∫ 2γ lv / RdS in the interior of the S 14 ACS Paragon Plus Environment

Page 15 of 41

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

liquid-solid interface due to the capillary pressure. During droplet departure process (maroon color droplet), the shifting of liquid-vapor interface results in a smaller dynamic contact angle θd 0 .

(8)

According to the calculation in our previous publication36, excess surface energy and surface adhesion work is proportional to the square of droplet radius R, while viscous dissipation is correlated to R1.5. So the criterion for droplet jumping, as a function of droplet size and contact angle θ, becomes

ξ sγ lv − ξad γ lv (1 + cos θ ) − ξvis µ γ lv / R > 0 ,

(9)

where ξs, ξad and ξvis are constant (see section S3 of the Supporting Information). From the equation, we can find that the role of viscous dissipation, though limited, increases with decreasing droplet radius. Therefore, it becomes difficult for the small droplet to retain enough kinetic energy to jump away. The ending points of the curves predicted by LBM simulation (in Fig. 8b) follow the inertial-capillary scaling law without any surface adhesion (contact angle =180°). In Fig. 8b, the curve for Oh=0.2 has the largest slope over the shortest contact angle range. It explains why the measured jumping velocities of small size droplets have a wide range on actual surfaces. The variation in the contact angle of a superhydrophobic surface can be due to nonuniform surface coating, nonuniform distribution of surface nano/microstructures and the existence of surface structures with different dimensions. In addition, different jumping velocity trends (shown in Fig. 8c) can be found for different surface hydrophobicity levels. At lower contact angles (green color region in Fig. 8b), the jumping velocity increases with the droplet 18 ACS Paragon Plus Environment

Page 19 of 41

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

size shown as the green curve in Fig. 8c. At higher contact angles (orange color region), the jumping velocity decreases with the droplet size. In the blue color region, the jumping velocity is in transition, as it rises to a peak then decreases with increasing droplet size. All these trends have been reported in previous literature.1-4 The minimum droplet size for jumping also depends on the surface hydrophobicity. We plot a phase map (Fig. 9) of jumping (solid symbols) and non-jumping events (empty symbols) jointly determined by droplet size and surface hydrophobicity. The black curve shows the theoretical prediction by Cha et al.44 The x-axis is the static contact angle without considering the contact angle hysteresis (CAH). According to the theoretical modelling by Cha et al.,44 there is a 10% mismatch compared with the ideal surface without CAH, if the influence of the CAH and the wetting behavior on the minimum droplet jumping size is considered. From the phase map, we can observe that smaller droplets can no longer jump while the larger ones can at a less hydrophobic surface. That is why some studies reported that droplets below 20 µm cannot jump on a moderate superhydrophobic surface,1 while some observed the jumping of droplets as small as 5 µm on a highly superhydrophobic surfaces.4 The jumping droplet size limit is decided by the surface hydrophobicity level.


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 41

FIG. 9. Jumping droplets phase map. The solid symbols represent successfully jumping droplets and the empty symbols represent droplets that fail to jump, predicted by the LBM simulations in our work. The solid curve is the boundary between the successful (colored) and unsuccessful jumping regimes predicted in reference.44

As mentioned at the very beginning, superhydrophobic surfaces are mostly nano-/micro structured. It should be noticed that the effect of surface structure size is not separately discussed above as it is assumed to be much smaller than the size of the droplets and thus integrated into the surface wettability. However, it is important to consider surface topography. It is essential to reduce the dimensions of surface structures, meanwhile ensuring the structures are uniformly distributed and homogenously chemically treated, in order to obtain a controllable and predictable droplet jumping performance.

3. CONCLUSION In this paper, we look into several recently reported studies on the droplet coalescence jumping. In order to probe the insight into the discrepancies found in these reports, we analyze the coalescence-jumping dynamics of small-sized (1–10 µm or Oh = 0.2–0.05) droplets on surfaces with various hydrophobicity levels. Three effects of the surface hydrophobicity, coupled with the droplet size, on the jumping velocity are demonstrated. First, the jumping velocity of a smaller droplet, larger Oh number, is found to be more sensitive to the surface hydrophobicity change. A slight change in the contact angle leads to a large change in the jumping velocity. It is also found that the surface hydrophobicity affects the trend of the jumping velocity change with the droplet 20 ACS Paragon Plus Environment

Page 21 of 41

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

size in a complicated manner. Specifically, we identify the range of surface hydrophobicity where the jumping velocity monotonically decreases with the droplet size as in most cases (θ≳170°), or increases (θ ≲160°), or changes non-monotonically (160°≲θ ≲170°). This is why the jumping velocity varies a lot in the small-droplet size region below 10 µm as reported by various researchers, while the jumping velocity of large droplets exhibits similar values for their respective surfaces with different hydrophobicity levels. Moreover, a phase diagram is generated from direct LBM simulation to show the minimum size of droplet jumping for different contact angles. These new insights on droplet jumping reveal the detailed physical mechanism behind various experimental observations, and contribute to the guidelines for designing superhydrophobic surfaces with enhanced droplet removal capability.

Supporting Information Additional information on the simulation method, observation of the oscillation during droplet coalescence, and the theoretical analysis of adhesion work and viscous dissipation

ACKNOWLEDGEMENT This work was supported by the Abu Dhabi National Oil Company Gas Sub Committee (ADCOC-GSC). The authors appreciate the assistance from Prof. Haibo Huang at the University of Science and Technology of China.


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 41

REFERENCES (1)

Boreyko, J. B.; Chen, C. H. Self-Propelled Dropwise Condensate on Superhydrophobic Surfaces. Phys. Rev. Lett. 2009, 103 (18), 2–5.

(2)

Enright, R.; Miljkovic, N.; Sprittles, J.; Nolan, K.; Mitchell, R.; Wang, E. N. How Coalescing Droplets Jump. ACS Nano 2014, 8 (10), 10352–10362.

(3)

Kim, M. K.; Cha, H.; Birbarah, P.; Chavan, S.; Zhong, C.; Xu, Y.; Miljkovic, N. Enhanced Jumping-Droplet Departure. Langmuir 2015, 31 (49), 13452–13466.

(4)

Cha, H.; Chun, J. M.; Sotelo, J.; Miljkovic, N. Focal Plane Shift Imaging for the Analysis of Dynamic Wetting Processes. ACS Nano 2016, No. 10, 8223–8232.

(5)

Aili, A.; Li, H.; Alhosani, M. H.; Zhang, T. Unidirectional Fast Growth and Forced Jumping of Stretched Droplets on Nanostructured Microporous Surfaces. ACS Appl. Mater. Interfaces 2016, 8 (33), 21776–21786.

(6)

Rykaczewski, K.; Paxson, A. T.; Anand, S.; Chen, X.; Wang, Z.; Varanasi, K. K. Multimode Multidrop Serial Coalescence Effects during Condensation on Hierarchical Superhydrophobic Surfaces. Langmuir 2013, 29 (3), 881–891.

(7)

Dietz, C.; Rykaczewski, K.; Fedorov, A. G.; Joshi, Y. Visualization of Droplet Departure on a Superhydrophobic Surface and Implications to Heat Transfer Enhancement during Dropwise Condensation. Appl. Phys. Lett. 2010, 97 (3), 1–4.

(8)

Miljkovic, N.; Wang, E. N. Condensation Heat Transfer on Superhydrophobic Surfaces. MRS Bull. 2013, 38 (5), 397–406.

(9)

Hou, Y.; Yu, M.; Chen, X.; Wang, Z.; Yao, S. Recurrent Filmwise and Dropwise 22 ACS Paragon Plus Environment

Page 23 of 41

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Condensation on a Beetle Mimetic Surface. ACS Nano 2015, 9 (1), 71–81. (10)

Miljkovic, N.; Enright, R.; Nam, Y.; Lopez, K.; Dou, N.; Sack, J.; Wang, E. N. JumpingDroplet-Enhanced Condensation on Scalable Superhydrophobic Nanostructured Surfaces. Nano Lett. 2013, 13 (1), 179–187.

(11)

Zhu, J.; Luo, Y.; Tian, J.; Li, J.; Gao, X. Clustered Ribbed-Nanoneedle Structured Copper Surfaces with High-Efficiency Dropwise Condensation Heat Transfer Performance. ACS Appl. Mater. Interfaces 2015, 7 (20), 10660–10665.

(12)

Attinger, D.; Frankiewicz, C.; Betz, A. R.; Schutzius, T. M.; Ganguly, R.; Das, A.; Kim, C.-J.; Megaridis, C. M. Surface Engineering for Phase Change Heat Transfer: A Review. MRS Energy Sustain. 2014, 1, E4.

(13)

Cho, H. J.; Preston, D. J.; Zhu, Y.; Wang, E. N. Nanoengineered Materials for Liquid– vapour Phase-Change Heat Transfer. Nat. Rev. Mater. 2016, 2, 16092.

(14)

Enright, R.; Miljkovic, N.; Alvarado, J. L.; Kim, K.; Rose, J. W. Dropwise Condensation on Micro- and Nanostructured Surfaces. Nanoscale Microscale Thermophys. Eng. 2014, 18 (3), 223–250.

(15)

Miljkovic, N.; Enright, R.; Wang, E. N. Effect of Droplet Morphology on Growth Dynamics and Heat Transfer during Condensation on Superhydrophobic Nanostructured Surfaces. ACS Nano 2012, 6 (2), 1776–1785.

(16)

Wisdom, K. M.; Watson, J. A.; Qu, X.; Liu, F.; Watson, G. S.; Chen, C.-H. Self-Cleaning of Superhydrophobic Surfaces by Self-Propelled Jumping Condensate. Proc. Natl. Acad. Sci. U. S. A. 2013, 110 (20), 7992–7997.


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(17)

Page 24 of 41

Watson, G. S.; Gellender, M.; Watson, J. A. Self-Propulsion of Dew Drops on Lotus Leaves: A Potential Mechanism for Self Cleaning. Biofouling 2014, 30 (4), 427–434.

(18) Zhang, Q.; He, M.; Chen, J.; Wang, J.; Song, Y.; Jiang, L.; Wang, E. N.; Wang, J.; Cao, Z.; Song, Y.; et al. Anti-Icing Surfaces Based on Enhanced Self-Propelled Jumping of Condensed Water Microdroplets. Chem. Commun. 2013, 49 (40), 4516. (19)

Boreyko, J. B.; Collier, C. P. Delayed Frost Growth on Jumping-Drop Superhydrophobic Surfaces. ACS Nano 2013, 7 (2), 1618–1627.

(20)

Zhang, K.; Liu, F.; Williams, A. J.; Qu, X.; Feng, J. J.; Chen, C.-H. Self-Propelled Droplet Removal from Hydrophobic Fiber-Based Coalescers. Phys. Rev. Lett. 2015, 115 (7), 74502.

(21)

Miljkovic, N.; Preston, D. J.; Enright, R.; Wang, E. N. Jumping-Droplet Electrostatic Energy Harvesting. Appl. Phys. Lett. 2014, 105 (1), 13111.

(22)

Tuteja, A.; Choi, W.; McKinley, G. H.; Cohen, R. E.; Rubner, M. F. Design Parameters for Superhydrophobicity and Superoleophobicity. MRS Bull. 2008, 33 (8), 752–758.

(23)

Crawford,

R.

J.

E.

P.

I.

Superdydrophobic

Surfaces,

Chapter:

Design

of

Surfacehydrophobic Surfaces, pp27-49. (24)

Raza, A.; Si, Y.; Ding, B.; Yu, J.; Sun, G. Fabrication of Superhydrophobic Films with Robust Adhesion and Dual Pinning State via in Situ Polymerization. J. Colloid Interface Sci. 2013, 395 (1), 256–262.

(25)

Sarkar, A.;

Kietzig,

A.-M. Design of a Robust Superhydrophobic Surface:

Thermodynamic and Kinetic Analysis. Soft Matter 2015, 11 (18), 3733–3733.


Page 25 of 41

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

(26)

Hao, C.; Liu, Y.; Chen, X.; Li, J.; Zhang, M.; Zhao, Y.; Wang, Z. Bioinspired Interfacial Materials with Enhanced Drop Mobility: From Fundamentals to Multifunctional Applications. Small 2016, 12 (14), 1825–1839.

(27)

Feng, B. L.; Li, S. H.; Li, Y. S.; Li, H. J.; Zhang, L. J.; Zhai, J.; Song, Y. L.; Liu, B. Q.; Jiang, L.; Feng, L.; et al. Super-Hydrophobic Surfaces: From Natural to Artificial. Adv. Mater. 2002, 14 (24), 1857–1860.

(28)

Liu, H.; Raza, A.; Aili, A.; Lu, J.; AlGhaferi, A.; Zhang, T. Sunlight-Sensitive AntiFouling Nanostructured TiO2 Coated Cu Meshes for Ultrafast Oily Water Treatment. Sci. Rep. 2016, 6 (April), 25414.

(29)

Rykaczewski, K.; Osborn, W. a.; Chinn, J.; Walker, M. L.; Scott, J. H. J.; Jones, W.; Hao, C.; Yao, S.; Wang, Z. How Nanorough Is Rough Enough to Make a Surface Superhydrophobic during Water Condensation? Soft Matter 2012, 8 (33), 8786.

(30)

Zeyu, M.; Hong, Y.; Liyuan, M.; Su, M. Superhydrophobic Membranes with Ordered Arrays of Nanospiked Microchannels for Water Desalination. Langmuir 2009, 25 (10), 5446–5450.

(31)

Barthwal, S.; Kim, Y. S.; Lim, S. H. Mechanically Robust Superamphiphobic Aluminum Surface with Nanopore-Embedded Microtexture. Langmuir 2013, 29 (38), 11966–11974.

(32)

Wu, X.; Bai, H.; Zhang, J.; Chen, F.; Shi, G. Copper Hydroxide Nanoneedle and Nanotube Arrays Fabricated by Anodization of Copper. J. Phys. Chem. B 2005, 109 (48), 22836–22842.

(33)

Zhao, Y.; Luo, Y.; Li, J.; Yin, F.; Zhu, J.; Gao, X. Condensate Microdrop Self-Propelling


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 41

Aluminum Surfaces Based on Controllable Fabrication of Alumina Rod-Capped Nanopores. ACS Appl. Mater. Interfaces 2015, 7 (21), 11079–11082. (34)

Gu, C.; Ren, H.; Tu, J.; Zhang, T. Y. Micro/nanobinary Structure of Silver Films on Copper Alloys with Stable Water-Repellent Property under Dynamic Conditions. Langmuir 2009, 25 (20), 12299–12307.

(35)

Tavana, H.; Amirfazli, A.; Neumann, A. W. Fabrication of Superhydrophobic Surfaces of N -Hexatriacontane. Langmuir 2006, 22 (13), 5556–5559.

(36)

Li, G.; Alhosani, M. H.; Yuan, S.; Liu, H.; Ghaferi, A. Al. Microscopic Droplet Formation and Energy Transport Analysis of Condensation on Scalable Superhydrophobic Nanostructured Copper Oxide Surfaces. Langmuir 2014, 30 (30), 14498.

(37)

Enright, R.; Miljkovic, N.; Al-Obeidi, A.; Thompson, C. V.; Wang, E. N. Condensation on Superhydrophobic Surfaces: The Role of Local Energy Barriers and Structure Length Scale. Langmuir 2012, 28 (40), 14424–14432.

(38)

Raj, R.; Enright, R.; Zhu, Y.; Adera, S.; Wang, E. N. Unified Model for Contact Angle Hysteresis on Heterogeneous and Superhydrophobic Surfaces. Langmuir 2012, 28 (45), 15777–15788.

(39)

Wang, F. C.; Yang, F.; Zhao, Y. P. Size Effect on the Coalescence-Induced Self-Propelled Droplet. Appl. Phys. Lett. 2011, 98 (5).

(40)

Liu, X.; Cheng, P. 3D Multiphase Lattice Boltzmann Simulations for Morphological Effects on Self-Propelled Jumping of Droplets on Textured Superhydrophobic Surfaces. Int. Commun. Heat Mass Transf. 2015, 64, 7–13.


Page 27 of 41

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

(41)

Liu, X.; Cheng, P.; Quan, X. Lattice Boltzmann Simulations for Self-Propelled Jumping of Droplets after Coalescence on a Superhydrophobic Surface. Int. J. Heat Mass Transf. 2014, 73 (March), 195–200.

(42)

Liu, F.; Ghigliotti, G.; Feng, J. J.; Chen, C.-H. Numerical Simulations of Self-Propelled Jumping upon Drop Coalescence on Non-Wetting Surfaces. J. Fluid Mech. 2014, 752, 39– 65.

(43)

Liu, F.; Ghigliotti, G.; Feng, J. J.; Chen, C.-H. Self-Propelled Jumping upon Drop Coalescence on Leidenfrost Surfaces. J. Fluid Mech. 2014, 752, 22–38.

(44)

Cha, H.; Xu, C.; Sotelo, J.; Chun, J. M.; Yokoyama, Y.; Enright, R.; Miljkovic, N. Coalescence-Induced Nanodroplet Jumping. Phys. Rev. Fluids 2016, 1 (6), 64102.

(45)

Liang, Z.; Keblinski, P. Coalescence-Induced Jumping of Nanoscale Droplets on SuperHydrophobic Surfaces. Appl. Phys. Lett. 2015, 107 (14).

(46)

Liu, T. Q.; Sun, W.; Sun, X. Y.; Ai, H. R. Mechanism Study of Condensed Drops Jumping on Super-Hydrophobic Surfaces. Colloids Surfaces A Physicochem. Eng. Asp. 2012, 414, 366–374.

(47)

Lv, C.; Hao, P.; Yao, Z.; Niu, F. Departure of Condensation Droplets on Superhydrophobic Surfaces. Langmuir 2015, 31 (8), 2414–2420.

(48)

Cheng, Y.; Xu, J.; Sui, Y. Numerical Investigation of Coalescence-Induced Droplet Jumping on Superhydrophobic Surfaces for Efficient Dropwise Condensation Heat Transfer. Int. J. Heat Mass Transf. 2016, 95, 506–516.

(49)

Peng, B.; Wang, S.; Lan, Z.; Xu, W.; Wen, R.; Ma, X. Analysis of Droplet Jumping


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 41

Phenomenon with Lattice Boltzmann Simulation of Droplet Coalescence. Appl. Phys. Lett. 2013, 102 (15). (50)

Gong, S.; Cheng, P. Numerical Investigation of Droplet Motion and Coalescence by an Improved Lattice Boltzmann Model for Phase Transitions and Multiphase Flows. Comput. Fluids 2012, 53 (1), 93–104.

(51)

Zhang, Q.; Sun, D.; Zhang, Y.; Zhu, M. Lattice Boltzmann Modeling of Droplet Condensation on Superhydrophobic Nanoarrays. Langmuir 2014, 30 (42), 12559–12569.

(52)

Lu, H. H. M. S. X. Multiphase Lattice Boltzmann Methods; 2015.

(53)

Shan, X.; Chen, H. Lattice Boltzmann Model for Simulating Flows with Multiple Phases and Components. Phys. Rev. E 1993, 47 (3), 1815–1819.

(54)

Huang, H.; Wang, L.; Lu, X. Y. Evaluation of Three Lattice Boltzmann Models for Multiphase Flows in Porous Media. Comput. Math. with Appl. 2011, 61 (12), 3606–3617.

(55)

Chen, L.; Kang, Q.; Mu, Y.; He, Y. L.; Tao, W. Q. A Critical Review of the Pseudopotential Multiphase Lattice Boltzmann Model: Methods and Applications. Int. J. Heat Mass Transf. 2014, 76, 210–236.

(56)

Yuan, P.; Schaefer, L. Equations of State in a Lattice Boltzmann Model. Phys. Fluids 2006, 18 (4).

(57)

Li, H.; Okaeme, C. C.; Yang, W.; Zhang, T. Effect of Surface Wettability and Gas/Liquid Velocity Ratio on Microscale Two-Phase Flow Patterns. In Proceedings of the ASME 2016 5th International Conference on Micro/Nanoscale Heat and Mass Transfer; ASME, 2016; p V002T10A002.


Page 29 of 41

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

(58)

Li, H.; Yang, W.; Huang, H.; Chevalier, S.; Sassi, M.; Zhang, T.; Zahaf, K.; Al-Farisi, O. Pore-Scale Lattice Boltzmann Simulation of Oil-Water Flow in Carbonate Rock with Variable Wettability. In Abu Dhabi International Petroleum Exhibition and Conference; Society of Petroleum Engineers, 2015; p SPE-177548-MS.

(59)

Huang, H.; Thorne, D. T.; Schaap, M. G.; Sukop, M. C. Proposed Approximation for Contact Angles in Shan-and-Chen-Type Multicomponent Multiphase Lattice Boltzmann Models. Phys. Rev. E - Stat. Nonlinear, Soft Matter Phys. 2007, 76 (6), 1–6.

(60)

Liu, Y.; Andrew, M.; Li, J.; Yeomans, J. M.; Wang, Z. Symmetry-Breaking in Drop Bouncing on Curved Surfaces. Nat. Commun. 2015, 6, 1–8.

(61)

Jung, Y. C.; Bhushan, B. Dynamic Effects of Bouncing Water Droplets on Superhydrophobic Surfaces. Langmuir 2008, 24 (12), 6262–6269.

(62)

Clanet, C.; Béguin, C.; Richard, D.; Quéré, D. Maximal Deformation of an Impacting Drop. J. Fluid Mech. 2004, 517, 199–208.

(63)

Moevius, L.; Liu, Y.; Wang, Z.; Yeomans, J. M. Pancake Bouncing: Simulations and Theory and Experimental Veri Fi Cation. Langmuir 2014, 30, 13021.

(64)

Weijs, J. H.; Andreotti, B.; Snoeijer, J. H. Elasto-Capillarity at the Nanoscale: On the Coupling between Elasticity and Surface Energy in Soft Solids. Soft Matter 2013, 9 (35), 8494.

(65)

Fernandez-Toledano, J.-C.; Blake, T. D.; Lambert, P.; De Coninck, J. On the Cohesion of Fluids and Their Adhesion to Solids: Young’s Equation at the Atomic Scale. Adv. Colloid Interface Sci. 2017, In Press. DOI: 10.1016/j.cis.2017.03.06


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(66)

Page 30 of 41

Dervaus, J.; Limat, L. Contact Lines on Soft Solids with Uniform Surface Tension: Analytical Solutions and Double Transition for Increasing Deformability. Proc. R. Soc. A 2014, 471 (20140813), R1.

(67)

Marchand, A.; Weijs, J. H.; Snoeijer, J. H.; Andreotti, B. Why Is Surface Tension a Force Parallel to the Interface? Am. J. Phys. 2011, 79 (October), 999.

(68)

Pericet-Camara, R.; Auernhammer, G. K.; Koynov, K.; Lorenzoni, S.; Raiteri, R.; Bonaccurso, E. Solid-Supported Thin Elastomer Films Deformed by Microdrops. Soft Matter 2009, 5 (19), 3611.


Page 31 of 41

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

TOC Graphic


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

FIG.1. Jumping velocity vs. droplet radius (reproduced from experimental results in references.1-4) 125x87mm (150 x 150 DPI)

ACS Paragon Plus Environment

Page 32 of 41

Page 33 of 41

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

FIG. 2. Momentum and streamlines at the early stage of coalescence jumping on a superhydrophobic surface. The lattice Boltzmann simulation result (bottom) is compared with finite-element-based simulation results from Ref.2 The color bar shows the momentum magnitude and direction, with the blue color representing positive (upward) and the yellow color representing negative (downward). 236x162mm (150 x 150 DPI)


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

FIG. 3. The evolution of the y-component velocity and droplet shape during the droplet coalescence jumping process simulated by LBM. The whole process is divided into three phases. In Phase I (coalescence), the surface curvature goes through a major evolution and the surface free energy is partially converted into kinetic energy. In Phase II (departure), the droplet overcomes the surface adhesion and departs from the solid surface. In Phase III, the drug force slows down the jumping droplet. 188x150mm (150 x 150 DPI)


Page 34 of 41

Page 35 of 41

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

FIG. 4. Contact area evolution during the coalescence jumping process. (a) Normalized contact area. At State S1, the contact area decreases first due to liquid flow from the droplet bottom towards the coalescence bridge. At S2, the liquid-vapor interface fully touches the solid surface. From S2 to S3, the downward flow bounces back and leads to the decrease of contact area. (b) Corresponding droplet morphologies and streamlines of the four states in (a). 805x586mm (144 x 144 DPI)


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

FIG. 5. Coalescence of two droplets pinned on needle tips. Oscillation of the liquid-vapor interface occurs due to the pinning effect and surface tension. The arrows are showing the liquid-vapor interface moving direction. 308x59mm (123 x 118 DPI)


Page 36 of 41

Page 37 of 41

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

FIG. 6. The effect of surface wettability on the jumping velocity of different droplet sizes. νj,s1 , νj,s2 , νj,l1 and νj,l2 indicate the jumping velocities of smaller (Oh = 0.2) and larger (Oh = 0.1) droplets with contact angles of 175° and 165°, respectively. When the contact angle changes from 175° to 165 °, the jumping velocity drop ∆νj,s of the smaller droplet (Oh = 0.2, dashed curve) is larger than that of the larger droplet (Oh = 0.1, solid curve). 134x94mm (150 x 150 DPI)


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

FIG. 7. Forces acting on a liquid droplet due to liquid-solid adhesion. At equilibrium state (blue colored droplet), in normal direction to the solid surface, the adhesion force exerted by solid-on-droplet at the contact line is balanced by the repelling force in the interior of the liquid-solid interface due to the capillary pressure. During droplet departure process (maroon color droplet), the shifting of liquid-vapor interface results in a smaller dynamic contact angle . Thus, the total force cannot become balanced, which results in a downward net force and causes the deceleration of νy+. 434x276mm (72 x 72 DPI)


Page 38 of 41

Page 39 of 41

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

FIG. 8. (a) The y+ maximum upward velocity νy+,max vs. the contact angle of water on the solid surface. The maximum velocity is not sensitive to the contact angle, but greatly affected by the droplet size. (b) The jumping velocity νj vs. contact angle. Red squares and blue circles are measured velocities for droplets Oh=0.1 and Oh=0.05 from different references1,2,4. (c) Three types of jumping velocity trends concluded from the jumping velocity νj vs. contact angle curves. 553x204mm (72 x 72 DPI)


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

FIG. 9. Jumping droplets phase map. The solid symbols represent successfully jumping droplets and the empty symbols represent droplets that fail to jump, predicted by the LBM simulations in our work. The solid curve is the boundary between the successful (colored) and unsuccessful jumping regimes predicted in reference.44 262x190mm (150 x 150 DPI)


Page 40 of 41

Page 41 of 41

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

TOC Graphic 392x202mm (72 x 72 DPI)


Insights into the Impact of Surface Hydrophobicity on Droplet

Recommend Documents